I was reading my favorite book on mathematics, Mathematical Problems and Proofs, in particular, a section on basic Set Theory. The book discusses the transitive relation, where if A is related to B, and B is related to C, then A is related to C. In this case, A, B, and C are abstract mathematical objects, but you can assign practical meaning by e.g., making them all integers, and considering ordinal relationships between them, where e.g., A is greater than B, B is greater than C, and therefore, A is greater than C.  Note that this example of ordinal relationships has a “therefore” clause, but relations are abstract statements of fact, not consequences of logic. That is, we simply posit relations between objects, whereas I’ve phrased the concrete example in terms of a logical conclusion, which is very different. That is, this example is consistent with the stated set of relations among A, B, and C, which are simply posited to exist, whereas the integers have properties that imply that A is greater than C as a matter of logic.

With that introduction, it dawned on me that we can consider higher order sets of relations that probably don’t have names like “transitive”. One obvious such set of relations is as follows, where A is related B, B is related to C, C is related to D, and A is related to D. All I did was add an extra object D, and extend the relations analogously. Specifically, we can express this as a graph, where A through D are connected by a path, and A is connected directly to D by an extra edge, creating what would be a circuit in an undirected graph. Though note that even if A is related to B, this does not imply that B is related to A, and as such, any graph expressing relations is directed. This is probably known, given how simple it is, and I’m certain through my own studies that people express relations using graphs.

The interesting bit is the possibility of using machines to discover meaningful higher order relations that e.g., require at least four or more objects. Because it’s at least possible for these relations to arise over any number of objects, we can’t give them all special names in a human language like “transitive”, but a machine can. The point being that, most of mathematics is probably accessible only to machines or other sentient beings capable of handling that much information, which plainly do not inhabit this planet in any appreciable number.